Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#1326 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #99 |
| commutator | C23 | C22×C8 |
| Frattini, cosocle | C2×C22⋊C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
385 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 385 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,17,18,15)(14,16);(12,15,16,17)(13,18)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1326, #128343, #128396, … |
| maximal normal subgroup of | C4×#1326, #128343, #128396, … |
| maximal quotient of | C2×#7064, #122316, #122320, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4439);